base-4.18.0.0: Basic libraries

Control.Arrow

Description

Basic arrow definitions, based on

• Generalising Monads to Arrows, by John Hughes, Science of Computer Programming 37, pp67-111, May 2000.

plus a couple of definitions (returnA and loop) from

• A New Notation for Arrows, by Ross Paterson, in ICFP 2001, Firenze, Italy, pp229-240.

Synopsis

# Arrows

class Category a => Arrow a where Source #

The basic arrow class.

Instances should satisfy the following laws:

• arr id = id
• arr (f >>> g) = arr f >>> arr g
• first (arr f) = arr (first f)
• first (f >>> g) = first f >>> first g
• first f >>> arr fst = arr fst >>> f
• first f >>> arr (id *** g) = arr (id *** g) >>> first f
• first (first f) >>> arr assoc = arr assoc >>> first f

where

assoc ((a,b),c) = (a,(b,c))

The other combinators have sensible default definitions, which may be overridden for efficiency.

Minimal complete definition

arr, (first | (***))

Methods

arr :: (b -> c) -> a b c Source #

Lift a function to an arrow.

first :: a b c -> a (b, d) (c, d) Source #

Send the first component of the input through the argument arrow, and copy the rest unchanged to the output.

second :: a b c -> a (d, b) (d, c) Source #

A mirror image of first.

The default definition may be overridden with a more efficient version if desired.

(***) :: a b c -> a b' c' -> a (b, b') (c, c') infixr 3 Source #

Split the input between the two argument arrows and combine their output. Note that this is in general not a functor.

The default definition may be overridden with a more efficient version if desired.

(&&&) :: a b c -> a b c' -> a b (c, c') infixr 3 Source #

Fanout: send the input to both argument arrows and combine their output.

The default definition may be overridden with a more efficient version if desired.

#### Instances

Instances details
 Monad m => Arrow (Kleisli m) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow Methodsarr :: (b -> c) -> Kleisli m b c Source #first :: Kleisli m b c -> Kleisli m (b, d) (c, d) Source #second :: Kleisli m b c -> Kleisli m (d, b) (d, c) Source #(***) :: Kleisli m b c -> Kleisli m b' c' -> Kleisli m (b, b') (c, c') Source #(&&&) :: Kleisli m b c -> Kleisli m b c' -> Kleisli m b (c, c') Source # Arrow (->) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow Methodsarr :: (b -> c) -> b -> c Source #first :: (b -> c) -> (b, d) -> (c, d) Source #second :: (b -> c) -> (d, b) -> (d, c) Source #(***) :: (b -> c) -> (b' -> c') -> (b, b') -> (c, c') Source #(&&&) :: (b -> c) -> (b -> c') -> b -> (c, c') Source #

newtype Kleisli m a b Source #

Constructors

 Kleisli FieldsrunKleisli :: a -> m b

#### Instances

Instances details
 Monad m => Category (Kleisli m :: Type -> Type -> Type) Source # Since: base-3.0 Instance detailsDefined in Control.Arrow Methodsid :: forall (a :: k). Kleisli m a a Source #(.) :: forall (b :: k) (c :: k) (a :: k). Kleisli m b c -> Kleisli m a b -> Kleisli m a c Source # Generic1 (Kleisli m a :: Type -> Type) Source # Instance detailsDefined in Control.Arrow Associated Typestype Rep1 (Kleisli m a) :: k -> Type Source # Methodsfrom1 :: forall (a0 :: k). Kleisli m a a0 -> Rep1 (Kleisli m a) a0 Source #to1 :: forall (a0 :: k). Rep1 (Kleisli m a) a0 -> Kleisli m a a0 Source # Monad m => Arrow (Kleisli m) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow Methodsarr :: (b -> c) -> Kleisli m b c Source #first :: Kleisli m b c -> Kleisli m (b, d) (c, d) Source #second :: Kleisli m b c -> Kleisli m (d, b) (d, c) Source #(***) :: Kleisli m b c -> Kleisli m b' c' -> Kleisli m (b, b') (c, c') Source #(&&&) :: Kleisli m b c -> Kleisli m b c' -> Kleisli m b (c, c') Source # Monad m => ArrowApply (Kleisli m) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow Methodsapp :: Kleisli m (Kleisli m b c, b) c Source # Monad m => ArrowChoice (Kleisli m) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow Methodsleft :: Kleisli m b c -> Kleisli m (Either b d) (Either c d) Source #right :: Kleisli m b c -> Kleisli m (Either d b) (Either d c) Source #(+++) :: Kleisli m b c -> Kleisli m b' c' -> Kleisli m (Either b b') (Either c c') Source #(|||) :: Kleisli m b d -> Kleisli m c d -> Kleisli m (Either b c) d Source # MonadFix m => ArrowLoop (Kleisli m) Source # Beware that for many monads (those for which the >>= operation is strict) this instance will not satisfy the right-tightening law required by the ArrowLoop class.Since: base-2.1 Instance detailsDefined in Control.Arrow Methodsloop :: Kleisli m (b, d) (c, d) -> Kleisli m b c Source # MonadPlus m => ArrowPlus (Kleisli m) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow Methods(<+>) :: Kleisli m b c -> Kleisli m b c -> Kleisli m b c Source # MonadPlus m => ArrowZero (Kleisli m) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow MethodszeroArrow :: Kleisli m b c Source # Alternative m => Alternative (Kleisli m a) Source # Since: base-4.14.0.0 Instance detailsDefined in Control.Arrow Methodsempty :: Kleisli m a a0 Source #(<|>) :: Kleisli m a a0 -> Kleisli m a a0 -> Kleisli m a a0 Source #some :: Kleisli m a a0 -> Kleisli m a [a0] Source #many :: Kleisli m a a0 -> Kleisli m a [a0] Source # Applicative m => Applicative (Kleisli m a) Source # Since: base-4.14.0.0 Instance detailsDefined in Control.Arrow Methodspure :: a0 -> Kleisli m a a0 Source #(<*>) :: Kleisli m a (a0 -> b) -> Kleisli m a a0 -> Kleisli m a b Source #liftA2 :: (a0 -> b -> c) -> Kleisli m a a0 -> Kleisli m a b -> Kleisli m a c Source #(*>) :: Kleisli m a a0 -> Kleisli m a b -> Kleisli m a b Source #(<*) :: Kleisli m a a0 -> Kleisli m a b -> Kleisli m a a0 Source # Functor m => Functor (Kleisli m a) Source # Since: base-4.14.0.0 Instance detailsDefined in Control.Arrow Methodsfmap :: (a0 -> b) -> Kleisli m a a0 -> Kleisli m a b Source #(<$) :: a0 -> Kleisli m a b -> Kleisli m a a0 Source # Monad m => Monad (Kleisli m a) Source # Since: base-4.14.0.0 Instance detailsDefined in Control.Arrow Methods(>>=) :: Kleisli m a a0 -> (a0 -> Kleisli m a b) -> Kleisli m a b Source #(>>) :: Kleisli m a a0 -> Kleisli m a b -> Kleisli m a b Source #return :: a0 -> Kleisli m a a0 Source # MonadPlus m => MonadPlus (Kleisli m a) Source # Since: base-4.14.0.0 Instance detailsDefined in Control.Arrow Methodsmzero :: Kleisli m a a0 Source #mplus :: Kleisli m a a0 -> Kleisli m a a0 -> Kleisli m a a0 Source # Generic (Kleisli m a b) Source # Instance detailsDefined in Control.Arrow Associated Typestype Rep (Kleisli m a b) :: Type -> Type Source # Methodsfrom :: Kleisli m a b -> Rep (Kleisli m a b) x Source #to :: Rep (Kleisli m a b) x -> Kleisli m a b Source # type Rep1 (Kleisli m a :: Type -> Type) Source # Since: base-4.14.0.0 Instance detailsDefined in Control.Arrow type Rep1 (Kleisli m a :: Type -> Type) = D1 ('MetaData "Kleisli" "Control.Arrow" "base" 'True) (C1 ('MetaCons "Kleisli" 'PrefixI 'True) (S1 ('MetaSel ('Just "runKleisli") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) ((FUN 'Many a :: Type -> Type) :.: Rec1 m))) type Rep (Kleisli m a b) Source # Since: base-4.14.0.0 Instance detailsDefined in Control.Arrow type Rep (Kleisli m a b) = D1 ('MetaData "Kleisli" "Control.Arrow" "base" 'True) (C1 ('MetaCons "Kleisli" 'PrefixI 'True) (S1 ('MetaSel ('Just "runKleisli") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (a -> m b)))) ## Derived combinators returnA :: Arrow a => a b b Source # The identity arrow, which plays the role of return in arrow notation. (^>>) :: Arrow a => (b -> c) -> a c d -> a b d infixr 1 Source # Precomposition with a pure function. (>>^) :: Arrow a => a b c -> (c -> d) -> a b d infixr 1 Source # Postcomposition with a pure function. (>>>) :: Category cat => cat a b -> cat b c -> cat a c infixr 1 Source # Left-to-right composition (<<<) :: Category cat => cat b c -> cat a b -> cat a c infixr 1 Source # Right-to-left composition ## Right-to-left variants (<<^) :: Arrow a => a c d -> (b -> c) -> a b d infixr 1 Source # Precomposition with a pure function (right-to-left variant). (^<<) :: Arrow a => (c -> d) -> a b c -> a b d infixr 1 Source # Postcomposition with a pure function (right-to-left variant). # Monoid operations class Arrow a => ArrowZero a where Source # Methods zeroArrow :: a b c Source # #### Instances Instances details  MonadPlus m => ArrowZero (Kleisli m) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow MethodszeroArrow :: Kleisli m b c Source # class ArrowZero a => ArrowPlus a where Source # A monoid on arrows. Methods (<+>) :: a b c -> a b c -> a b c infixr 5 Source # An associative operation with identity zeroArrow. #### Instances Instances details  MonadPlus m => ArrowPlus (Kleisli m) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow Methods(<+>) :: Kleisli m b c -> Kleisli m b c -> Kleisli m b c Source # # Conditionals class Arrow a => ArrowChoice a where Source # Choice, for arrows that support it. This class underlies the if and case constructs in arrow notation. Instances should satisfy the following laws: • left (arr f) = arr (left f) • left (f >>> g) = left f >>> left g • f >>> arr Left = arr Left >>> left f • left f >>> arr (id +++ g) = arr (id +++ g) >>> left f • left (left f) >>> arr assocsum = arr assocsum >>> left f where assocsum (Left (Left x)) = Left x assocsum (Left (Right y)) = Right (Left y) assocsum (Right z) = Right (Right z) The other combinators have sensible default definitions, which may be overridden for efficiency. Minimal complete definition (left | (+++)) Methods left :: a b c -> a (Either b d) (Either c d) Source # Feed marked inputs through the argument arrow, passing the rest through unchanged to the output. right :: a b c -> a (Either d b) (Either d c) Source # A mirror image of left. The default definition may be overridden with a more efficient version if desired. (+++) :: a b c -> a b' c' -> a (Either b b') (Either c c') infixr 2 Source # Split the input between the two argument arrows, retagging and merging their outputs. Note that this is in general not a functor. The default definition may be overridden with a more efficient version if desired. (|||) :: a b d -> a c d -> a (Either b c) d infixr 2 Source # Fanin: Split the input between the two argument arrows and merge their outputs. The default definition may be overridden with a more efficient version if desired. #### Instances Instances details  Monad m => ArrowChoice (Kleisli m) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow Methodsleft :: Kleisli m b c -> Kleisli m (Either b d) (Either c d) Source #right :: Kleisli m b c -> Kleisli m (Either d b) (Either d c) Source #(+++) :: Kleisli m b c -> Kleisli m b' c' -> Kleisli m (Either b b') (Either c c') Source #(|||) :: Kleisli m b d -> Kleisli m c d -> Kleisli m (Either b c) d Source # ArrowChoice (->) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow Methodsleft :: (b -> c) -> Either b d -> Either c d Source #right :: (b -> c) -> Either d b -> Either d c Source #(+++) :: (b -> c) -> (b' -> c') -> Either b b' -> Either c c' Source #(|||) :: (b -> d) -> (c -> d) -> Either b c -> d Source # # Arrow application class Arrow a => ArrowApply a where Source # Some arrows allow application of arrow inputs to other inputs. Instances should satisfy the following laws: • first (arr (\x -> arr (\y -> (x,y)))) >>> app = id • first (arr (g >>>)) >>> app = second g >>> app • first (arr (>>> h)) >>> app = app >>> h Such arrows are equivalent to monads (see ArrowMonad). Methods app :: a (a b c, b) c Source # #### Instances Instances details  Monad m => ArrowApply (Kleisli m) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow Methodsapp :: Kleisli m (Kleisli m b c, b) c Source # ArrowApply (->) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow Methodsapp :: (b -> c, b) -> c Source # newtype ArrowMonad a b Source # The ArrowApply class is equivalent to Monad: any monad gives rise to a Kleisli arrow, and any instance of ArrowApply defines a monad. Constructors  ArrowMonad (a () b) #### Instances Instances details  Source # Since: base-4.6.0.0 Instance detailsDefined in Control.Arrow Methodsempty :: ArrowMonad a a0 Source #(<|>) :: ArrowMonad a a0 -> ArrowMonad a a0 -> ArrowMonad a a0 Source #some :: ArrowMonad a a0 -> ArrowMonad a [a0] Source #many :: ArrowMonad a a0 -> ArrowMonad a [a0] Source # Arrow a => Applicative (ArrowMonad a) Source # Since: base-4.6.0.0 Instance detailsDefined in Control.Arrow Methodspure :: a0 -> ArrowMonad a a0 Source #(<*>) :: ArrowMonad a (a0 -> b) -> ArrowMonad a a0 -> ArrowMonad a b Source #liftA2 :: (a0 -> b -> c) -> ArrowMonad a a0 -> ArrowMonad a b -> ArrowMonad a c Source #(*>) :: ArrowMonad a a0 -> ArrowMonad a b -> ArrowMonad a b Source #(<*) :: ArrowMonad a a0 -> ArrowMonad a b -> ArrowMonad a a0 Source # Arrow a => Functor (ArrowMonad a) Source # Since: base-4.6.0.0 Instance detailsDefined in Control.Arrow Methodsfmap :: (a0 -> b) -> ArrowMonad a a0 -> ArrowMonad a b Source #(<$) :: a0 -> ArrowMonad a b -> ArrowMonad a a0 Source # ArrowApply a => Monad (ArrowMonad a) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow Methods(>>=) :: ArrowMonad a a0 -> (a0 -> ArrowMonad a b) -> ArrowMonad a b Source #(>>) :: ArrowMonad a a0 -> ArrowMonad a b -> ArrowMonad a b Source #return :: a0 -> ArrowMonad a a0 Source # (ArrowApply a, ArrowPlus a) => MonadPlus (ArrowMonad a) Source # Since: base-4.6.0.0 Instance detailsDefined in Control.Arrow Methodsmzero :: ArrowMonad a a0 Source #mplus :: ArrowMonad a a0 -> ArrowMonad a a0 -> ArrowMonad a a0 Source #

leftApp :: ArrowApply a => a b c -> a (Either b d) (Either c d) Source #

Any instance of ArrowApply can be made into an instance of ArrowChoice by defining left = leftApp.

# Feedback

class Arrow a => ArrowLoop a where Source #

The loop operator expresses computations in which an output value is fed back as input, although the computation occurs only once. It underlies the rec value recursion construct in arrow notation. loop should satisfy the following laws:

extension
loop (arr f) = arr (\ b -> fst (fix (\ (c,d) -> f (b,d))))
left tightening
loop (first h >>> f) = h >>> loop f
right tightening
loop (f >>> first h) = loop f >>> h
sliding
loop (f >>> arr (id *** k)) = loop (arr (id *** k) >>> f)
vanishing
loop (loop f) = loop (arr unassoc >>> f >>> arr assoc)
superposing
second (loop f) = loop (arr assoc >>> second f >>> arr unassoc)

where

assoc ((a,b),c) = (a,(b,c))
unassoc (a,(b,c)) = ((a,b),c)

Methods

loop :: a (b, d) (c, d) -> a b c Source #

#### Instances

Instances details
 MonadFix m => ArrowLoop (Kleisli m) Source # Beware that for many monads (those for which the >>= operation is strict) this instance will not satisfy the right-tightening law required by the ArrowLoop class.Since: base-2.1 Instance detailsDefined in Control.Arrow Methodsloop :: Kleisli m (b, d) (c, d) -> Kleisli m b c Source # ArrowLoop (->) Source # Since: base-2.1 Instance detailsDefined in Control.Arrow Methodsloop :: ((b, d) -> (c, d)) -> b -> c Source #